Advances in Operator Theory

$T1$ theorem for inhomogeneous Triebel–Lizorkin and Besov spaces on RD-spaces and its application

Fanghui Liao, Zongguang Liu, and Hongbin Wang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Using Calderón's reproducing formulas and almost orthogonal estimates, the $T1$ theorem for the inhomogeneous Triebel–Lizorkin and Besov spaces on RD-spaces is obtained. As an application, new characterizations for these spaces with “half” the usual conditions of the approximate to the identity are presented.

Article information

Source
Adv. Oper. Theory Volume 3, Number 3 (2018), 522-537.

Dates
Received: 22 September 2017
Accepted: 28 January 2018
First available in Project Euclid: 7 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1518016383

Digital Object Identifier
doi:10.15352/aot.1709-1236

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B35: Function spaces arising in harmonic analysis

Keywords
$T1$ theorem Triebel–Lizorkin space Besov space RD-space

Citation

Liao, Fanghui; Liu, Zongguang; Wang, Hongbin. $T1$ theorem for inhomogeneous Triebel–Lizorkin and Besov spaces on RD-spaces and its application. Adv. Oper. Theory 3 (2018), no. 3, 522--537. doi:10.15352/aot.1709-1236. https://projecteuclid.org/euclid.aot/1518016383


Export citation

References

  • R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer-Verlag, Berlin, 1971.
  • G. David and J.-L. Journé, A boundedness criterion for generalized Calderón–Zygmund operators, Ann. of Math. 120 (1984), no. 2, 371–397.
  • G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón–Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 1–56.
  • L. Grafakos, L. Liu, and D. Yang, Vector-valued singular integrals and maximal functions on spaces of homogeneous type, Math. Scand. 104 (2009), no. 2, 296–310.
  • Y. C. Han, New characterizations of inhomogeneous Besov and Triebel–Lizorkin spaces over spaces of homogeneous type, Acta Math. Sin. (Engl. Ser.) 25 (2009), no. 11, 1787–1804.
  • Y. S. Han, D. Müller, and D. Yang, Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr. 279 (2006), no. 13-14, 1505–1537.
  • Y. S. Han, D. Müller, and D. Yang, A theory of Besov and Triebel–Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces, Abstr. Appl. Anal. 2008, Art. ID 893409, 250 pp.
  • Y. S. Han, S. Lu, and D. Yang, Inhomogeneous Besov and Triebel–Lizorkin spaces on spaces of homogeneous type, Approx. Theory Appl. 15 (1999), no. 3, 37–65.
  • Y. S. Han and E. Sawyer, Littlewood-Paley theory on the spaces of homogeneous type and classical function spaces, Mem. Amer. Math. Soc. 110 (1994), no. 530, 1–126.
  • Y. C. Han and Y. Xu, New characterizations of Besov and Triebel–Lizorkin spaces over spaces of homogeneous type, J. Math. Anal. Appl. 325 (2007), no. 1, 305–318.
  • F. Liao, Y. C. Han, and Z. Liu, New characterizations of Besov and Triebel–Lizorkin spaces via the $T1$ theorem, Illinois J. Math. 60 (2016), no. 2, 391–412.
  • Y. Meyer, Wavelets and operators, Translated from the 1990 French original by D. H. Salinger. Cambridge Studies in Advanced Mathematics, 37. Cambridge University Press, Cambridge, 1992.
  • Y. Meyer and R. R. Coifman, Wavelets, Calderón–Zygmund and multilinear operators, Translated from the 1990 and 1991 French originals by David Salinger. Cambridge Studies in Advanced Mathematics, 48. Cambridge University Press, Cambridge, 1997.
  • R. A. Macías and C. Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257–270.
  • A. Nagel and E. M. Stein, On the product theory of singular integrals, Rev. Mat. Iberoameriicana 20 (2004), no. 2, 531–561.
  • D. Yang, $T1$ Theorem on Besov and Triebel–Lizorkin spaces on spaces of homogeneous type and their applications, Z. Anal. Anwendungen 22 (2003), no. 1, 53–72.
  • D. Yang, Real interpolations for Besov and Triebel–Lizorkin spaces on spaces of homogeneous type, Math. Nachr. 273(2004), no. 1, 96–113.
  • D. Yang and Y. Zhou, New properties of Besov and Triebel–Lizorkin spaces on RD-spaces, Manuscripta Math. 134 (2011), no. 1-2, 59–90.